Page 67 - Applied Numerical Methods Using MATLAB

56 MATLAB USAGE AND COMPUTATIONAL ERRORS
vector of the integral polynomial for a given polynomial coefficient
vector.
(cf) MATLAB has the built-in routines polyder()/polyint() for finding the
derivative/integral of a polynomial.

function pd = poly_der(p)
%p: the vector of polynomial coefficients in descending order
N = length(p);
if N <= 1, pd = 0; % constant
else
fori=1:N- 1, pd(i) = p(i)*(N - i); end
end

(f) Roots of A Polynomial Equation: roots()
Write a MATLAB statement to get the roots of the following polynomial
equation
8
p(x) = x − 1 = 0 (P1.11.5)
You can check if the result is right, by using the MATLAB command
poly(), which generates a polynomial having a given set of roots.
(g) Partial Fraction Expansion of a Ratio of Two Polynomials: residue()/
residuez()
(i) The MATLAB routine [r,p,k] = residue(B,A) finds the partial
fraction expansion for a ratio of given polynomials B(s)/A(s) as

B(s) b 1 s M−1 + b 2 s M−2 +· · · + b M r(i)
= = k(s) +
A(s) a 1 s N−1 + a 2 s N−2 + ··· + a N s − p(i)
i
(P1.11.6a)
which is good for taking the inverse Laplace transform. Use this
routine to find the partial fraction expansion for
4s + 2
X(s) = = + +
3
2
s + 6s + 11s + 6 s + s + s +
(P1.11.7a)
(ii) The MATLAB routine [r,p,k] = residuez(B,A) finds the par-
tial fraction expansion for a ratio of given polynomials B(z)/A(z)
as

B(z) b 1 + b 2 z −1 +· · · + b M z −(M−1) −1 r(i)z
= = k(z ) +
A(z) a 1 + a 2 z −1 +· · · + a N z −(N−1) z − p(i)
i
(P1.11.6b)
which is good for taking the inverse z-transform. Use this routine
to find the partial fraction expansion for
4 + 2z −1 z z z
X(z) = = + +
1 + 6z −1 + 11z −2 + 6z −3 z + z + z +
(P1.11.7b)